College Physics

3 Two-Dimensional Kinematics

The arc of a basketball, the orbit of a satellite, a bicycle rounding a curve, a swimmer diving into a pool, blood gushing out of a wound, and a puppy

chasing its tail are but a few examples of motions along curved paths. In fact, most motions in nature follow curved paths rather than straight lines.

Motion along a curved path on a flat surface or a plane (such as that of a ball on a pool table or a skater on an ice rink) is two-dimensional, and thus

described by two-dimensional kinematics. Motion not confined to a plane, such as a car following a winding mountain road, is described by three-

dimensional kinematics. Both two- and three-dimensional kinematics are simple extensions of the one-dimensional kinematics developed for straight-

line motion in the previous chapter. This simple extension will allow us to apply physics to many more situations, and it will also yield unexpected

insights about nature.

3.1 Kinematics in Two Dimensions: An Introduction

Figure 3.2Walkers and drivers in a city like New York are rarely able to travel in straight lines to reach their destinations. Instead, they must follow roads and sidewalks,

making two-dimensional, zigzagged paths. (credit: Margaret W. Carruthers)

Two-Dimensional Motion: Walking in a City

Suppose you want to walk from one point to another in a city with uniform square blocks, as pictured inFigure 3.3.

Figure 3.3A pedestrian walks a two-dimensional path between two points in a city. In this scene, all blocks are square and are the same size.

The straight-line path that a helicopter might fly is blocked to you as a pedestrian, and so you are forced to take a two-dimensional path, such as the

one shown. You walk 14 blocks in all, 9 east followed by 5 north. What is the straight-line distance?

An old adage states that the shortest distance between two points is a straight line. The two legs of the trip and the straight-line path form a right

triangle, and so the Pythagorean theorem,a^2 + b^2 = c^2 , can be used to find the straight-line distance.

Figure 3.4The Pythagorean theorem relates the length of the legs of a right triangle, labeledaandb, with the hypotenuse, labeledc. The relationship is given by:

a^2 + b^2 = c^2. This can be rewritten, solving forc:c = a^2 + b^2.

The hypotenuse of the triangle is the straight-line path, and so in this case its length in units of city blocks is

(9 blocks)^2 + (5 blocks)^2 = 10.3 blocks, considerably shorter than the 14 blocks you walked. (Note that we are using three significant figures in

86 CHAPTER 3 | TWO-DIMENSIONAL KINEMATICS

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